Sewall Wright's adaptive topography is a central paradigm in evolutionary biology, demonstrating that, under certain assumptions, gene frequency evolution in a constant environment maximizes the mean fitness of individuals in a population (Wright 1932, 1937, 1969). On Wright's adaptive topography a population is located at a point on the surface of mean fitness as a function of gene frequencies. Assuming genotypic fitnesses constant in time, with random mating and approximate linkage equilibrium among loci, natural selection in a constant environment causes gene frequencies to change such that the population moves uphill on the adaptive topography, increasing the mean fitness in the population. Fisher expressed a similar concept in his “fundamental theorem of natural selection,” in which the rate of increase of the mean Malthusian fitness, or population growth rate, equals the additive genetic variance in fitness (Fisher 1930, 1958 chap. 2; interpretations by different authors reviewed by Crow 2002). Wright's initial intuitive explanations of the adaptive topography, and his shifting balance theory of evolution (Wright 1931, 1932), were later clarified by explicit models demonstrating multiple peaks on the adaptive topography, and showing that gene frequency change by selection depends on the gradient of mean fitness with respect to gene frequencies (Wright 1935a,b, 1937, 1939, 1956, 1969).

Adaptive topographies have been criticized, most notably by Fisher (1941) and Moran (1964) who produced examples of evolution in which mean fitness either does not change or decreases. Fisher's example concerned a plant population in which an allele causing self-fertilization (without reducing pollen output or seed production) becomes rapidly fixed. This entails that genotypic fitnesses are not constant but change with the genotypic frequencies (Porcher and Lande 2005). Moran's example concerned a negative epistatic interaction between two polymorphic loci in a population initially in complete linkage disequilibrium, containing only the two most-fit genotypes, which by random mating and recombination then produce less-fit genotypes, reducing the mean fitness without changing gene frequencies. These examples respectively violated Wright's assumptions of random mating with constant genotypic fitnesses, and approximate linkage equilibrium. Wright (1942, 1949) showed that under nonrandom mating, or with frequency-dependent fitnesses, evolution generally does not maximize mean fitness. Kimura (1965) developed the concept of quasi-linkage equilibrium, demonstrating that when the strength of epistatic selection is much less than the recombination rate, random mating quickly produces a nearly constant linkage disequilibrium after which gene frequency evolution increases the mean fitness. The main criticisms were reviewed and answered by Wright (1967, 1988) and Crow and Kimura (1970, pp. 195–236).

Wright's adaptive topography therefore remains an important conceptual tool for analysis and understanding of evolution in both simple and complex genetic systems (e.g., Simpson 1953; Futuyma 1998; Gavrilets 2004). An analogous adaptive topography was derived for evolution of the mean phenotypes of quantitative characters, influenced by multiple genes and environmental effects, assuming constant phenotypic fitnesses and (multivariate) normal distributions of phenotypes and additive genetic values, with approximately constant phenotypic and additive genetic variances (and covariances) (Lande 1976, 1979, 1982).

The theory reviewed above concerns evolution in a constant environment. However, all populations experience temporally fluctuating environments, causing population size and natural selection to fluctuate. Analyzing empirical time series of selection coefficients, using existing methods for measuring selection, is a natural way of describing fluctuating selection. A population in a fluctuating environment is conceived as evolving toward a continually shifting optimum, with the mean fitness often temporarily decreasing (Crow and Kimura 1970, p. 236; Gillespie 1991, p. 305). Environmental change is one component of Fisher's (1958, pp. 44–45) “deterioration of the environment.” This raises the question: Is there a general principle of adaptation in a fluctuating environment?

A related question in stochastic demography concerns how to quantify the long-run growth rate of population size under density-independent growth in a fluctuating environment. Assuming a stationary distribution of environmental states, fundamental demographic results have been obtained using the concept of the “long-run growth rate” defined as the expected rate of increase of ln *N*, the natural logarithm of population size (Cohen 1977, 1979). In general, for populations with overlapping generations, the long-run growth rate can be approximated as , in which *r* is the population growth rate in the average environment, and σ^{2}_{e} is the environmental variance in population growth rate (Tuljapurkar 1982). The term long-run growth rate arises from the following consideration. In the absence of density dependence, the stochastic process for ln *N* has constant incremental mean and variance per unit time, describing Brownian motion with a deterministic trend (Lande and Orzack 1988). Starting from ln *N*(0) at time *t*= 0, at a later time the distribution of ln *N*(*t*) is asymptotically normal with mean and variance σ^{2}_{e}*t*. The slope of a population trajectory on the log scale, [ln *N*(*t*) − ln *N*(0)]/*t*, then has mean and variance σ^{2}_{e}/*t*. With increasing time the variance of the slope approaches zero and all populations eventually have the same slope, (Cohen 1977, 1979; Tuljapurkar 1982; Caswell 2001; Lande et al. 2003).

Early models of fluctuating selection on a single genetic locus in a population with discrete nonoverlapping generations showed that the final outcome of selection is determined by the geometric mean fitness of a genotype through time (Dempster 1955; Haldane and Jayakar 1963; Felsenstein 1976; Gillespie 1973, 1991, p. 147). The analog of the geometric mean fitness in continuous time is the long-run growth rate, which is also applicable with overlapping generations (Tuljapurkar 1982; Lande and Orzack 1988; Caswell 2001; Lande et al. 2003). For alleles at a single haploid locus (or an asexual population) the genotype with the highest long-run growth rate will eventually become fixed in the population (Gillespie 1991, p. 147). At a diploid locus, the long-run growth rate of a rare heterozygote determines whether it can successfully invade a large population in a fluctuating environment (Tuljapurkar 1982; Gillespie 1991; Charlesworth 1994), and a permanent fluctuating polymorphism of two alleles can be maintained if the long-run growth rate of the heterozygote exceeds that of either heterozygote (Gillespie 1973, 1991). At a multiallelic diploid locus, the long-run growth rates of all the genotypes determine the eventual outcome of weak fluctuating selection (Turelli 1981). However, before accepting the geometric mean fitness or the long-run growth rate as the expected fitness of a genotype (Gillespie 1973, 1977; Tuljapurkar 1982, 1990; Metz et al. 1992; Caswell 2001, chap. 14.6; Doak et al. 2005), note that the above conclusions concern the eventual fate of a gene, and not the expected change in gene frequency over a short period of time, by which the expected selection coefficients and expected fitnesses of genotypes should be defined.

Here I analyze a well-known model of fluctuating selection on a haploid locus, and develop an analogous model of fluctuating selection on quantitative characters. In both models the expected evolution is governed by the pattern of genetic variation and the gradient of the long-run growth rate of the population, , with respect to gene frequency, *p*, or mean phenotype, . This shows that the expected evolution increases . The long-run growth rate therefore provides an adaptive topography determining the expected selection, which does not change with time despite environmental fluctuations. The population, represented as a point on this surface, is expected to evolve as a stochastic maximization of . In a haploid population this principle is not realized because the expected optimizing selection is dominated by stochasticity. For a quantitative character a stochastic maximization of can occur under fluctuating selection if the variance of the standardized selection gradient (in units of phenotypic standard deviations) is small or the heritability is low, which often will be satisfied. The adaptive surface as a function of , therefore provides an intuitive guide to phenotypic evolution in a fluctuating environment.

The analysis also reveals the form of the expected relative fitness of a genotype or phenotype within a population. Based on the results of simple genetic models mentioned above, prevailing opinion plausibly suggests that genotypic or phenotypic fitnesses should be measured by long-run growth rates. In contrast, for both the haploid (asexual) and quantitative character models of fluctuating selection, I find that the expected relative fitness of a genotype or phenotype is its Malthusian fitness in the average environment minus the covariance between its growth rate and that of the population.